fast and curious

Saturday Teaser Curio #70: Impossible Football

Are you ready for some (American) football? The NFL playoffs are in full swing, which, of course, makes you want to do brain teasers. Right?

In regular American football, teams can score 2 points for a safety and 3 points for a field goal. In the right combinations, any point total > 1 is possible. But in this imaginary new-fangled who-cares-about-defense version of American football, teams can only score 3 points for a field goal or 7 points for a touchdown (extra points are automatic). No other scoring plays exist.

What's the largest score that is impossible for a team to get?

Ready for the answer?

Did you know this actually wasn't a football question at all? This is a problem in mathematics called the Frobenius coin problem (follow the link to geek out!).

You can see how it's related to coins; what amount of money is impossible to make given certain denominations? In this case, imagine a country with only two denominations: $3 and $7. What is the largest amount of money that is impossible to have?

Here, the answer is 11. There is no combination of 3 and 7 that can make 11.

How do we know we can score any number of points 12 or higher?

12 = 3 + 3 +3 +3

13 = 7 + 3 + 3

14 = 7 + 7

When we can just add additional 3s to each of these to make any score we want.